On generating set for cubic hypersurfaces over $\mathbb{Q}$ of high dimension

We will prove that for sufficiently large $n$ on every smooth cubic hypersurface $X \subset \PP^n (\mathbb{Q})$ there exists a point $P$ which generates all others through secant and tangent constructions. More precisely, the following holds: $P = S_0 \subset S_1 \subset \ldots \subset S_m = X$, where $S_i \setminus S_{i-1}$ consists of the points on $X$ through which such a line can be drawn that its other intersections with $X$ lie in $S_{i-1}$. We'll need the statement of the theorem by Skinner on weak approximation. This theorem is the main reason for dimension to be bounded from below.